Koopman operator and its approximations for systems with symmetries

Abstract

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces block diagonal structure in operator approximations, revealing hidden organization. Practically, if the data is symmetric, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise.

Sparsity structures for Koopman operators

“…The optimization problem (28) has the advantage that its solution gives an optimal invariant measure while ( 29) respectively (40) only provide (pseudo) moments of the marginals of such a measure. That is why, in case one is interested in the extremal measure and not only in the optimal value or its marginals, we suggest using (28) and to exploit its block structure.…”

“…The optimization problem (28) has the advantage that its solution gives an optimal invariant measure while ( 29) respectively (40) only provide (pseudo) moments of the marginals of such a measure. That is why, in case one is interested in the extremal measure and not only in the optimal value or its marginals, we suggest using (28) and to exploit its block structure. Another way is to solve (29) by means of solving (40) and extend the obtained marginals to an (invariant) measure by solving a feasibility problem for linear matrix inequalities, i.e.…”

“…In terms of exploiting structures our work is related to [28] where it is shown that symmetry translates to operators commuting with the Koopman operators. In this text we show that the considered sparse structures allow to intertwine the Koopman operator T of the whole system with the Koopman operators T I for corresponding smaller systems, i.e.…”

“…Both the approaches, via symmetry [28] and sparsity in this text, formulate properties of the dynamical system as properties of the Koopman operator. This follows the classical idea of Koopman theory (see for example [16]), namely to translate properties of f or its corresponding dynamical system into functional analytic properties of the Koopman operator -and vice versa.…”

Local Koopman Operators for Data-Driven Control of Robotic Systems

“…There has not been a systematic way to address general nonlinear systems; rather, most efforts rely on trial-and-error [40]- [42], [44]- [46] and machine learning tools [39], [47], or are system-specific [26]. Furthermore, there is no method available to bound the modeling error of the finite-dimensional Koopman operators for general nonlinear systems.…”